Question

A Gulab Jamun when completely ready for eating contains sugar syrup up to about 30 % of its volume. Find approximately how much syrup would be found in 45 Gulab Jamun shaped like a cylinder with two hemispherical ends if the complete length of each of the Gulab Jamun is 5 cm and its diameter is 2.8 cm.

Answer 1

Hint: In this question, to find the volume of the sugar syrup, we first have to find the volume of Gulab Jamun and the volume of the Gulab Jamun can be found by adding the volumes of three parts: cylinder and 2 hemispheres.

Complete step-by-step solution -
A Gulab Jamun is shaped like a cylinder with two hemispherical ends.

Therefore,
The volume of the Gulab Jamun = Volume of the Cylinder + Volume of 2 hemispheres
We know that, the diameter of the cylinder = $2.8 \text{cm}$.
Therefore,
\[\text{Radius}=\dfrac{\text{Diameter}}{2}=\dfrac{2.8}{2}=1.4cm\]
Here, the diameter of the hemisphere = $2.8 \text{cm}$. Therefore, radius = $1.4 \text{cm}$.
Now, the total length of the Gulab Jamun = Height of the cylinder + 2 x Radius of the hemisphere.
\[\Rightarrow 5={h_c}+2\times 1.4\]
\[\Rightarrow {h_c}=5-2\times 1.4\]
\[\Rightarrow {h_c}=5-2.8=2.2cm\]
Therefore, the height of the cylinder is $2.2 \text{cm}$.
Therefore, the volume of the cylinder = \[\pi {{r}^{2}}h\]
\[=\dfrac{22}{7}\times {{\left( 1.4 \right)}^{2}}\times 2.2\]
\[=22\times 0.2\times 1.4\times 2.2\]
\[=22\times 0.616\]
\[=13.55c{{m}^{3}}\]
The volume of the hemisphere \[=\dfrac{2}{3}\pi {{r}^{3}}\]
\[=\dfrac{2}{3}\times \dfrac{22}{7}\times {{\left( 1.4 \right)}^{3}}\]
\[=\dfrac{2}{3}\times 22\times 0.2\times 1.4\times 1.4\]
\[=5.749\]
\[=5.75c{{m}^{3}}\]
Therefore, the volume of the 2 hemispheres = 2 x Volume of the hemisphere
\[=2\times 5.75\]
\[=11.50c{{m}^{3}}\]
Therefore, the volume of the Gulab Jamun
$= 13.55 + 11.50$
\[=25.05c{{m}^{3}}\]
Therefore, the volume of 1 Gulab Jamun \[=25.05c{{m}^{3}}\]
Therefore, the volume of 45 Gulab Jamun \[=45\times 25.05c{{m}^{3}}\]
We need to find the quantity of the syrup in 45 Gulab Jamuns. It is given that the quantity of the syrup in 1 Gulab Jamun is 30 %.
Therefore, the quantity of the syrup in 45 Gulab Jamuns
\[=30\% \times 45\times 25.05\]
\[=\dfrac{30}{100}\times 45\times 25.05\]
\[=\dfrac{27}{2}\times 25.05\]
\[=27\times 12.525\]
\[=338.175c{{m}^{3}}\]
\[=338c{{m}^{3}}\left( \text{approx} \right)\]
Therefore, 45 Gulab Jamuns contain \[338c{{m}^{3}}\] of syrup.

Note: Here, remember that to get the volume of the syrup in 45 Gulab Jamuns, we first have to find the total volume of one Gulab Jamun. After doing so, can either calculate the volume of the syrup in one Gulab Jamun and then multiply it by 45 or calculate the total volume of 45 Gulab Jamuns and then calculate the volume of the syrup in them. Either way, you will get the same answer.